The red indicator shows that the chat is new. Usually, within the indicator, there will be a small number, showing the number of messages the visitor has sent. Once the agent clicks on the chat and the field to type in an answer, the number will disappear. Once the answer is sent to the customer, the red indicator will disappear as well, since the chat is no longer new and it is being replied to.
The next new chat will come after the current new chat. Once the first new chat is being replied to, they will switch places as the red indicator is the most important one and the chats that are with that indicator are on the top of the line.